Multiplicative Error Analysis: A Scale-Invariant View of TDDFT Two-Photon Absorption Benchmarks

Two-photon absorption (2PA) strengths span several orders of magnitude, yet time-dependent density functional theory (TDDFT) benchmark studies have commonly relied on raw-scale mean absolute errors and linear regression statistics. Such statistics can be dominated by the most intense transitions and do not distinguish systematic multiplicative bias, typical fold error, dynamic range reproduction, and rank preservation. Here, published TDDFT 2PA benchmarks based on coupled-cluster references are reanalyzed by using a scale-invariant, multiplicative framework. For the most widely used benchmark set [J. Chem.Theory Comput. 2022, 18, 1046], it is shown that the five strongest transitions, representing only approximately 10% of the data, account for up to 65.5% of the total raw absolute error. Consequently, raw-scale metrics assign disproportionate weight to a small number of intense transitions and may obscure method performance across the remainder of the intensity range. Across several benchmark sets, the analysis further reveals that similar raw-scale errors can arise from qualitatively different error structures. The multiplicative analysis goes beyond previously recognized qualitative trends by resolving them into directly comparable and interpretable components. These results demonstrate that neither individual raw-scale metrics nor their combinations provide a sufficient basis for identifying a reliable or generally recommended functional, and they support a multidimensional, scale-invariant assessment of 2PA benchmarks. The same framework may also be useful for other properties for which multiplicative errors provide a more appropriate description than additive deviations such as oscillator strengths, kinetic and radiative rate constants, and equilibrium constants.